Alexander Chichigin
THE AUTHOR SPECIFICALLY DISCLAIMS ANY WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
\[ F(x, y, y', \dots, y^{(n-1)}) = y^{(n)} \]
\[ y' = F(x, y) \]
\[ y_{n+1} = y_n + h \times F(x_n, y_n) \]
\[ \overrightarrow{y}_{n+1} = F_n \left( \mathbf{W}_n\times \overrightarrow{y}_n + \overrightarrow{b}_n \right) \]
where
\[ F_n \in \left\{ \sigma, ~\mathrm{ReLU}, ~\dots \right\} \]
\[ \overrightarrow{y}_{n+1} = \overrightarrow{y}_n + F_n \left( \mathbf{W}_n\times \overrightarrow{y}_n + \overrightarrow{b}_n \right) \]
\[ \overrightarrow{y}(t + dt) = \overrightarrow{y}(t) + dt \times G \left( \overrightarrow{y}(t), t, \theta \right) \]
\[ \frac{ d\overrightarrow{y}(t) }{dt} = G \left( \overrightarrow{y}(t), t, \theta \right) \]
\[ \frac{ d\mathbf{y}(t)}{dt} = G \left( \mathbf{y}(t), t, \theta \right) \]
\[ \mathbf{y}(t_1) = \mathtt{ODESolve}(\mathbf{y}(t_0), G, t_0, t_1, \theta) \]
\[ L(\mathbf{y}(t_1)) = L(\mathtt{ODESolve}(\mathbf{y}(t_0), G, t_0, t_1, \theta)) \]
\[ u' = f(u, x) \] and \[ u(0) = u_0 \]
\[ u(x) \approx NN_{\theta}(x) \]
Ideally \[ \frac{d NN_{\theta}}{dx}(x) = f(NN_{\theta}(x), x) \]
\[ L(\theta) = \sum_i \left( \frac{d NN_{\theta}}{dx}(x_i) - f(NN_{\theta}(x_i), x_i) \right)^2 \]
\[ g_{\theta}(x) = u_0 + x \times NN_{\theta}(x) \]
\[ L(\theta) = \sum_i \left( \frac{d g_{\theta}}{dx}(x_i) - f(g_{\theta}(x_i), x_i) \right)^2 \]
\[ u_t + \mathcal{N}[u] = 0, x \in \Omega, t \in [0, T] \]
\[ f := u_t + \mathcal{N}[u] \]
\[ u(t, x) \approx NN_{\theta}(t, x) \]
\[ MSE = MSE_u + MSE_f \] where
\[ MSE_u = \frac{1}{N_u} \sum_{i=1}^{N_u} \left( u(t_u^i, x_u^i) - u^i \right)^2 \] and
\[ MSE_f = \frac{1}{N_f} \sum_{i=1}^{N_f} \left( f(t_f^i, x_f^i) \right)^2 \]
\[ u_t + \mathcal{N}[u; \lambda] = 0, x \in \Omega, t \in [0, T] \]
\[ f := u_t + \mathcal{N}[u; \lambda] \]
\[ u(t, x) \approx NN_{\theta}(t, x) \]
\[ \mathcal{N}[u(t), u(\alpha(t)), W(t), U_\theta(u, \beta(t))] = 0 \]
where \(\alpha(t)\) is a delay function and \(W(t)\) is the Wiener process.
I don’t really know, but…